The study of shallow water flow with bottom topography by high-order compact gas-kinetic scheme on unstructured mesh

Abstract

A well-balanced compact high-order gas-kinetic scheme (GKS) on unstructured mesh is first developed for solving the shallow water equations with source terms. The distinguishable feature of the finite volume GKS is that based on the gas-kinetic formulation, a time-accurate gas distribution function can be constructed, from which both the fluxes and the flow variables can be explicitly evaluated at the cell interface. As a result, besides the update of cell-averaged conservative variables, the cell-averaged slopes of the flow variables can be updated as well. Equipped with both flow variables and their slopes, a fourth-order compact spatial reconstruction on unstructured mesh can be obtained as the initial condition at the beginning of each time step. For the shallow water flow, in order to preserve the well-balanced property, the advection and the source terms in the flux function have to be balanced properly. The current compact GKS achieves high-order accuracy, keeps the well-balanced property, and has super-robustness in the simulation of bore waves. The scheme is used in the shallow water flow studies, such as dam breaking and bore wave propagation. In addition, the pollution transport, morphodynamics, and bottom friction in the shallow water flow have been included in the scheme. In the end, the water discharge in the Pearl River estuary and the dam-break experiment with movable bed topography have been simulated.